miniaturized loop resonator filter using …chapter 2 : microstrip resonators and slow wave...

MINIATURIZED LOOP RESONATOR FILTER USING

CAPACITIVELY LOADED TRANSMISSION LINES

MAK HON YEONG

DEPARTMENT OF ELECTRICAL AND

COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2005

MINIATURIZED LOOP RESONATOR FILTER USING

CAPACITIVELY LOADED TRANSMISSION LINES

MAK HON YEONG

B.Eng (Hons.), NUS

A THESIS SUBMITTED

FOR THE DEGREE OF MASTERS IN ENGINEERING

DEPARTMENT OF ELECTRICAL AND

COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2005

i

Table Of Contents

List of Tables .................................................................................................................iii

List of Figures ................................................................................................................ iv

Abstract ........................................................................................................................vii

Acknowledgements ...................................................................................................viii

Chapter 1 : Introduction............................................................................................... 1

1.1 Introduction..................................................................................................... 1

1.2 Objectives ....................................................................................................... 2

1.3 Scope of Work ................................................................................................ 2

1.4 Organization.................................................................................................... 3

1.5 Publications Arising from the Present Work .................................................. 4

Chapter 2 : Microstrip Resonators and Slow Wave Structures ................................... 5

2.1 Introduction..................................................................................................... 5

2.2 Microstrip Transmission Line......................................................................... 5

2.3 Microstrip Resonator ...................................................................................... 7

2.4 Ring Resonator................................................................................................ 9

2.4.1 Ring Equivalent Circuit and Input Impedance...................................... 11

2.4.2 Modes, Perturbations, and Coupling Methods of Ring Resonators...... 12

2.4.3 Applications Using Ring Resonators .................................................... 14

2.5 Slow Wave Structures................................................................................... 15

2.5.1 Lossless Transmission Line .................................................................. 15

2.5.2 Capacitive Loaded Transmission Lines (CTL)..................................... 16

Chapter 3 : Closed Loop Resonator Miniaturisation ................................................ 19

3.1 Introduction................................................................................................... 19

3.2 Novel Closed Loop Resonator ...................................................................... 20

3.3 Resonator Synthesis Procedure..................................................................... 25

3.3.1 Example ................................................................................................ 31

3.4 Summary ....................................................................................................... 38

Chapter 4 : Filter Synthesis Using Arbitrary Resonator Structures........................... 39

4.1 Band Pass Filters........................................................................................... 39

4.2 Coupled Resonator Filter .............................................................................. 40

4.2.1 General Coupling Matrix for Coupled Resonator Filters ..................... 40

4.2.2 General formulation for Extracting Coupling Coefficient K................ 43

4.2.3 Formulation for Extracting External Quality Factor Qe........................ 45

ii

4.3 Procedure for Coupled Resonator Filter Design........................................... 47

Chapter 5 : Loaded Q and Coupling Coefficient of Resonators................................ 48

5.1 Introduction................................................................................................... 48

5.2 Loaded Q of Resonators................................................................................ 48

5.2.1 Coupled Line Coupling......................................................................... 48

5.2.2 Tapped Line Coupling .......................................................................... 51

5.2.3 Other Explored Feed Structures............................................................ 55

5.3 Coupling Coefficient K of Resonators.......................................................... 58

5.4 Summary ....................................................................................................... 60

Chapter 6 : Miniaturized Closed Loop Resonator Filter ........................................... 61

6.1 Introduction................................................................................................... 61

6.2 Chebyshev Filter of 0.01dB Ripple, N=3, BW=10% Using Square Closed

Loop resonator .......................................................................................................... 61

6.3 Chebyshev Filter of 0.01dB ripple, BW=10% Using Resonator dbl_d66.... 64

6.4 Fabricated and Measured Results ................................................................. 66

6.5 Summary ....................................................................................................... 68

Chapter 7 : Conclusion .............................................................................................. 69

7.1 Suggestion for Future Works ........................................................................ 70

Chapter 8 : Appendix................................................................................................. 71

References..................................................................................................................... 80

iii

List of Tables

Table 1: QL of tapped line coupled structures for dbl_d66........................................... 54

Table 2: Coupling coefficient measurement results...................................................... 59

Table 3: Normalized K and Q values for Chebyshev filter 0.01dB ripple 10%

bandwidth ....................................................................................................... 61

iv

List of Figures

Figure 1: Microstrip Line................................................................................................ 5

Figure 2: Ring resonator ................................................................................................. 9

Figure 3: ring resonator with one feed line................................................................... 10

Figure 4: Equivalent circuit of ring resonator............................................................... 11

Figure 5: Maximum field points for different resonant modes.................................... 13

Figure 6: Ring resonator with slit ................................................................................. 13

Figure 7: Lossless transmission line circuit .................................................................. 15

Figure 8: Capacitively loaded transmission line.......................................................... 16

Figure 9: Square closed loop f0=1.42 GHz ................................................................... 22

Figure 10: Double_stub f0=1.19 GHz ........................................................................... 22

Figure 11: dbl_w35 f0=1.14 GHz................................................................................. 22

Figure 12: dbl_d66 f0=1.08 GHz ................................................................................. 22

Figure 13: Square closed loop at f0=1.08 GHz ............................................................. 22

Figure 14: Resonator frequency response..................................................................... 23

Figure 15: Compare resonance frequency .................................................................... 23

Figure 16: Single stub unit cell ..................................................................................... 28

Figure 17: Double stub unit cell.................................................................................... 28

Figure 18: Cascaded unit cells for a single side............................................................ 28

Figure 19: Circuit model of miniaturized resonator in ADS ........................................ 29

Figure 20: Resonator Synthesis Procedure ................................................................... 30

Figure 21: Double stub EM model................................................................................ 32

Figure 22: Double stub ADS circuit model .................................................................. 33

Figure 23: Double stub response................................................................................... 33

Figure 24: Single stub EM model ................................................................................ 34

Figure 25: Single stub circuit model............................................................................ 34

Figure 26: Single stub response ................................................................................... 35

Figure 27: Synthesized resonator.................................................................................. 36

Figure 28: Synthesized resonator dB|S21| .................................................................... 37

Figure 29: Equivalent circuit for n-coupled resonators (a) loop equation formulation,

(b) network representation ............................................................................................ 40

Figure 30: Singly loaded resonator S11........................................................................ 46

Figure 31: Parallel coupled line feed ............................................................................ 49

Figure 32: Coupled line with interdigital stubs feed..................................................... 49

v

Figure 33: Coupled line QL ........................................................................................... 49

Figure 34: Coupled line with X interdigital stubs (gap=15 mils) ................................. 50

Figure 35: QL vs no. of stubs ........................................................................................ 50

Figure 36: Tapped line coupling for dbl_d66 ............................................................... 51

Figure 37: Tapped line coupling for Square closed loop resonator .............................. 51

Figure 38: Tapped line with series inductor ................................................................. 51

Figure 39: Effect of series inductor on QL .................................................................... 52

Figure 40: Multiple tap feed structures......................................................................... 53

Figure 41: QL of tapped line coupled structures ........................................................... 54

Figure 42: Single tap with vertical shifted tap positions.............................................. 55

Figure 43: Single tap Q-loaded vs Offset ..................................................................... 56

Figure 44: Multiple tap with vertical shifted feed positions ........................................ 56

Figure 45: Multiple tap Q-loaded vs Offset .................................................................. 57

Figure 46: Coupling measurement of resonator dbl_d66 ............................................. 58

Figure 47: Coupling measurement of Square closed loop resonators........................... 58

Figure 48: Coupling Coefficient K vs Gap ................................................................... 59

Figure 49: Layout of the 3rd order Chebyshev filter using Square closed loop

resonator........................................................................................................................ 62

Figure 50: Circuit simulation of 3rd order Chebyshev filter using Square closed loop

resonator with series inductors placed at each port to increase QL............................... 63

Figure 51: Simulated response of the 3rd order filter using Square closed loop resonator

....................................................................................................................................... 63

Figure 52: Layout of the 3rd Order Chebyshev filter using dbl_d66............................ 65

Figure 53: Simulated response (IE3D) of the 3rd order filter using dbl_d66 ................ 65

Figure 54: Fabricated 3rd order filter using dbl_d66..................................................... 66

Figure 55: VNA measurement from 50 MHz to 2400 MHz......................................... 66

Figure 56: VNA measurement 50 MHz to 1600 MHz ................................................. 67

Figure 57: Compare simulated vs measured result ....................................................... 67

vi

List of Symbols

ε0 permittivity

µ0 permeability

λ wavelength

CTL capacitively loaded transmission line

TL transmission line

up phase velocity

vii

Abstract This thesis details the design and investigation of a miniaturized microstrip closed loop

resonator using slow wave structures in the form of capacitively loaded microstrip

lines. The primary objective is to achieve resonator and hence filter miniaturization

with a secondary objective of achieving improving resonator coupling to aid filter

synthesis.

A novel miniaturized closed loop resonator structure that achieves both miniaturization

and improved coupling has been developed. The miniaturized resonator is

demonstrated to achieve a 37% reduction in area when compared against a square

closed loop resonator of equivalent resonant frequency. Also developed are feed

structures to provide improved control of external QL. To aid resonator design, a

methodology to synthesize the new structure based on frequency requirements is

provided.

A 3rd order Chebyshev filter using the new resonator structure has been fabricated. In

comparison to a filter synthesized using a closed loop resonator of similar size, the

new structure achieves a 22% lower resonant frequency and also an additional area

reduction of 6% which is possible due to space savings provided by the structure.

viii

Acknowledgements I would like to thank Professor Leong Mook Seng, Associate Professor Ooi Ban Leong

and Doctor Chew Siou Teck for their invaluable advice and guidance to this project. I

would also like to thank the staff from the Radio Frequency Laboratory at DSO for

providing support for fabrication processes.

Lastly, I would like to thank Mr Ng Tiong Huat and friends at the Microwave

Laboratory of NUS for their company and friendship. They have made my academic

experience a fulfilling and enriching one.

1

Chapter 1 : Introduction

1.1 Introduction

The need for miniaturization in modern mobile communication systems has presented

new challenges to the design of high performance miniature RF filters and resonators.

This is especially so where the frequency of operation falls within L-band (1-2 GHz)

and S-band (2-4 GHz).

Microwave resonant structures are used extensively in applications such as filters,

oscillators and amplifiers. At low frequencies, resonant structures are realized using

lumped elements. At microwave frequencies the use of cavity and microstrip

resonators are commonly employed.

A microstrip resonator is any structure that is able to contain at least one oscillating

EM field [5]. In general, microstrip resonators can be classified as lump-element or

quasi-lumped element resonators and distributed line or patch resonators.

Various methods have been developed to achieve miniaturization, one of which is by

exploiting the slow wave effect using capacitively loaded transmission lines (CTL).

The CTL concept has been applied in various structures to reduce the size of planar

circuits [1] - [3].

This thesis focuses on planar microstrip closed loop resonator and filter design. It

proposes the use of CTL to reduce resonator size and concurrently increase coupling

between resonators by using stubs to form an interdigital capacitor structure.

2

1.2 Objectives

The objective of this thesis is to study novel methods to enable miniaturization of

microstrip closed loop resonators. The objectives can be listed as follows:

o To develop compact closed loop resonators by using capacitively loaded

transmission lines (CTL).

o To develop resonators capable of providing improved coupling performance

o To synthesize filters by using the newly developed resonators

1.3 Scope of Work

The scope of this project can be divided into 3 main portions. The first portion

explores the various types of Microstrip resonators.

The second portion looks into slow wave structures and explores how it can be applied

to Microstrip closed loop resonators to achieve miniaturisation. A miniaturised closed

loop resonator using slow wave structure is developed here. Also developed is a

methodology for resonator synthesis.

The third portion explores coupled resonator filter synthesis using the miniaturized

closed loop resonator. The new resonator structure is characterized for its design

curves such as external Q and coupling coefficient K. Using the characterized

information, a third order Chebyshev filter is successfully developed, fabricated and

tested.

3

1.4 Organization

Chapter 2 provides an overview of the types of planar microstrip resonators. A brief

description of the different types of microstrip resonators is provided. As the project is

focused on closed loop resonators, a common type namely the ring resonators is

explored in more detail.

Chapter 3 explores the principle behind slow wave structures and explains how it can

be applied to planar microstrip lines in the form of capacitively loaded transmission

lines (CTL). A method to synthesize CTL structures is also developed.

Chapter 4 proposes a new class of microstrip closed loop resonators which uses the

CTL to achieve resonator miniaturization. The newly synthesized structure is able to

achieve 22% reduction in frequency. A method to synthesize the new structure is also

developed.

Chapter 5 provides an overview of filter synthesis using arbitrary resonator structures

and explains coupled resonator filter synthesis.

Chapter 6 characterizes the external Q and coupling coefficient K of the newly

developed resonator structure. Feed structures for coupled line and tapped line

coupling are developed for the new resonator structure.

Chapter 7 performs filter synthesis using the newly developed resonator structure. A

3rd order Chebyshev filter of 0.01dB ripple and bandwidth 10% is synthesized here.

Chapter 8 concludes with discussions on the work done and results achieved.

Suggestions for future studies are proposed.

4

1.5 Publications Arising from the Present Work

Based on the works of this research, a paper has been submitted for review and

publication in the Microwave and Optical Technology Letters Journal:

H.Y. Mak, S.T. Chew, M.S. Leong, B.L. Ooi, “A Modified Miniaturized Loop

Resonator Filter”

5

Chapter 2 : Microstrip Resonators and Slow Wave Structures

2.1 Introduction

Microwave resonators are used in a variety of applications, including filters, oscillators,

frequency meters and tuned amplifiers. [6]

At microwave frequencies, distributed elements are commonly used to achieve

resonance. Resonators can be made using microstrip transmission lines, cavities and

dielectrics.

2.2 Microstrip Transmission Line

Various forms of planar transmission lines have been developed. Some examples are

strip line, Microstrip line, slot line and coplanar waveguide. The Microstrip line is the

most popular type and will be described here.

Figure 1: Microstrip Line

The geometry of a Microstrip line is shown Figure 1. A conducting strip of width W,

thickness t lies on the top of a substrate with dielectric constant εr and a thickness h. At

the bottom of the substrate lies a continuous conducting ground plane.

6

The Microstrip line is an inhomogeneous transmission line. The field between the strip

and the ground plane are not contained entirely in the substrate but extends within two

media, air and dielectric. Hence the microstrip line cannot support a pure TEM wave.

The mode of propagation is quasi-TEM.

The phase velocity and propagation constant can be expressed as

e

pcuε

= ek εβ 0= .

The effective dielectric constant of a microstrip line is given approximately by

Wdrr

e/121

12

12

1+

−+

+=

εεε .

The effective dielectric constant can be interpreted as the dielectric constant of a

homogeneous medium that replaces the air and dielectric regions of the microstrip.

Given the dimensions of the microstrip line, the characteristic impedance can be

calculated as

( )[ ]

≥+++

≤

+

=1for W/d

444.1/ln6697.0393.1/120

1for W/d 4

8ln60

e

0

dWdW

dW

Wd

Z e

επ

ε

For given characteristic impedance Z0 and dielectric constant εr the W/d ratio can be

found as

( )

>

−+−

−+−−−

<−

=2for W/d 61.039.0)1ln(

21

12ln12

2for W/d 2

82

rr

r

A

A

BBB

ee

dW

εεε

π

Where

+

+−

++

=rr

rrZAεε

εε 11.023.011

21

600

rZB

επ

02377

=

7

2.3 Microstrip Resonator

A microstrip resonator is any structure that is able to contain at least one oscillating

EM field [5]. There are many forms of microstrip resonators. In general, microstrip

resonators for may be classified as lumped element or quasi-lumped element

resonators and distributed line or patch resonators. A brief introduction to each of the

different resonator types will be explained below.

Lumped or quasi-lumped resonators will oscillate at LC

fπ2

10 = .

However they may resonate at other higher frequencies at which their sizes are no

longer much smaller than a wavelength. At those frequencies they will no longer

behave as lumped or quasi-lumped elements.

8

Distributed line resonators are formed by using microstrip lines of various wavelengths

(4λ ,

2λ , λ ) where λ is the guided wavelength at the fundamental resonant frequency fo.

The quarter wavelength resonator 4λ long resonates the fundamental frequency 0f and

at other frequencies of ( ) 02 1 for 2,3,f n f n= − = …

The half wavelength resonator 2λ long resonates at the fundamental frequency 0f and

at other frequencies of 0 for 2,3,f nf n= = … this type of resonator can also be

shaped into open-loop resonators.

The full wavelength resonator λ long resonates at the fundamental frequency 0f and at

other frequencies of 0 for 2,3,f nf n= = … this type of resonator is commonly

found in the form of ring or closed loop resonators with a median circumference

2 rπ λ= , where r is the radius of the ring. Because of its symmetrical geometry a

resonance can occur in either of 2 orthogonal coordinates. This type of line resonator

has a distinct feature; it can support a pair of degenerate modes that have the same

resonant frequencies but orthogonal field distributions. This feature can be utilised to

design dual mode filters.

Patch resonators provide increased power handling capability. An associated

advantage of patch resonators is their lower conductor losses as compared with narrow

microstrip line resonators. Patch resonators usually have a larger size; however, this

would not be a problem for applications in which the power handling or low loss is a

higher priority.

9

2.4 Ring Resonator

This project focuses on the development of closed loop resonators. A common closed

loop resonator type is the ring resonator. This consists of a transmission line formed in

a circular closed loop. The basic ring resonator circuit consists of feed lines, coupling

gaps, and the resonator (Figure 2). Power is coupled into and out of the resonator

through feed lines and coupling gaps.

Figure 2: Ring resonator

The ring resonator is a full wavelength resonator λ long. It resonates when the mean

circumference of the ring resonator is equal to an integral multiple of a guided

wavelength. This may be expressed as

2 , for n=1,2,3,...gr nπ λ= 02 eff

ncf nfrπ ε

= =

where geff

cf

λε

=

and r is the mean radius of the ring that equals the average of the outer and inner radii.

λg is the guided wavelength and n is the mode number. For the first mode, the maxima

of field occur at the coupling gap locations and nulls occur 90o from the coupling gap

locations. This relationship is only valid for the loose or weakly coupled case, as it

does not account for loading effects from the ports. [9]

10

Coupling is said to be weak or “loosely coupled” if the distance between the feed lines

and the resonator is large enough such that the resonant frequency of the ring is not

affected. If the feed lines are moved closer to the resonator, the gap capacitance

increases. If capacitance is sufficiently large, resonator loading will occur and may

cause the resonant frequency of the circuit to deviate from the intrinsic resonant

frequencies of the ring. Hence when measuring a resonator, the capacitance of the

coupling gaps has to be considered.

X

Figure 3: ring resonator with one feed line

The ring can be fed by using only one feed line Figure 3. This configuration is used in

dielectric constant, Q-measurements and ring stabilised oscillations. In this

configuration for the first mode, maximum field occurs at the coupling gap however a

minimum occurs at the opposite side 180o from the coupling gap. Hence when fed

using a single feed, the ring behaves as a half wavelength resonator. Resonance occurs

when the ring circumference is equals half of a guide wavelength:

2 , for n=1,2,3,...2

gr nλ

π = 4 eff

ncfrπ ε

= .

11

2.4.1 Ring Equivalent Circuit and Input Impedance

The ring resonator Figure 2 can be modelled by a lumped-parameter equivalent circuit

in the form a 2-port network Figure 4. The circuit can be reduced to a 1-port circuit by

terminating one of the ports with arbitrary impedance. The terminating impedance

should correspond with the feed impedance, which is usually 50 ohms. [11]

C1C1

C2

ZaZa ZbC1C1

C2

Za

Zb

Za

Figure 4: Equivalent circuit of ring resonator

Because of symmetry of the circuit, the input impedance can be found by simplifying

parallel and series combinations. The input impedance is expressed as:

( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

21 2 1 2 1 1 2

2 22 21 2 1 1 2 1 1 2

1 21 2 1 1 2

2 22 21 2 1 1 2 1 1 2

2

2 2

2

2 2

in

C C C C C D C C CR

C C D C C C C C C C

D C C C C C C

C C D C C C C C C C

ω

ω ω

ω ω

ω ω

−

+ + − + = + − + + +

+ − + + + − + + +

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

1 21 2 1 2 1 1 2

2 22 21 2 1 1 2 1 1 2

2 21 2 1 2 1 1 2

2 22 21 2 1 1 2 1 1 2

2

2 2

2

2 2

in

D C C C C D C C CX

C C D C C C C C C C

D C C C C C C C C

C C D C C C C C C C

ω ω

ω ω

ω

ω ω

− + − + − + = + − + + +

+ − + + + + − + + +

where

12

( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )

2

22

2

2 2

22

22 21 2 1 1 2

2 2 21 2 1 1 2 1 2

22 21 2 1 1 2

(2 ) 2

212 2 2

2

2

2

b

a b

b a ba b

a b

AZCA Z B Z

Z Z B ZD Z Z

A Z B Z

RCAC C R C C C

C C R C C C C CB

C C R C C C

ω

ω

ω ω ω

=+ − −

− −= − −

+ − −

= + + +

+ + + +=

+ + +

where R is the terminated load,

and the input impedance is in in inZ R jX= +

Resonant occurs when Xin=0.

2.4.2 Modes, Perturbations, and Coupling Methods of Ring Resonators

The ring resonator supports various different modes. The modes excited in the annular

ring element can be controlled by adjusting the excitation and perturbation. Resonant

modes are divided into groups according to types of excitation and perturbation. They

are the:

1) Regular mode, and

2) Forced mode

Regular resonant modes

A regular mode is obtained by applying symmetric input and output feedlines on the

annular ring element. The resonant wavelengths of the regular mode are determined

by 2 gr nπ λ= .[9]

The ring can be analysed as 2 half-wavelength linear resonators connected in parallel.

The parallel connection removes problems related to radiation from open ends hence

enabling a higher Q compared to linear resonator.

Resonance occurs when standing waves are setup in the ring, this happens when

circumference is integer multiple of guided wavelength. In the absence of gaps or other

discontinuities, maximum field occurs at the position where the feed line excites the

resonator. The number of maximum field points increases with the mode order as

shown in Figure 5.

13

n=1 XX n=2 XX

X

X

n=3 XX n=4 XX

X

X

X X

X X

X X

X X

Figure 5: Maximum field points for different resonant modes

Forced Resonant Modes

Forced modes are excited by forced boundary conditions on a microstrip annular ring

element. The boundary condition can either be open or short. The open boundary

condition is realised by cutting slits on the annular ring element. The shorted boundary

condition is realised by inserting vias to ground inside substrate. This forces minima of

electric field to occur on both sides of the shorted plane.

With the boundary conditions determined, the standing wave pattern and hence

maximum field points inside the ring can be determined.

Figure 6: Ring resonator with slit

14

2.4.3 Applications Using Ring Resonators

The microstrip ring resonator is applied in many different applications such as for

measurement applications, filters, couplers, magic-T circuits and antennas. A brief

summary of the many different applications for ring resonators are given below:

1) Measurement: Dispersion , dielectric constant ,

Q measurement and discontinuity measurements

2) Filter: bandpass filters, dual-mode bandpass filters,

Slotline ring filters

3) Ring couplers: 180° Rat-race hybrid ring couplers,

180° reverse phase back to back baluns,

180° reverse phase hybrid ring couplers,

90° branch line couplers

4) Ring Magic-T Circuits: 180° double-sided slotline ring magic T

5) Ring Antennas: Slotline ring antennas,

Dual frequency ring antennas

15

2.5 Slow Wave Structures

This sub-section explains the principle behind slow wave structures and how they can

be applied to Microstrip lines. First the Lossless transmission line will be explained.

Following that, the Capacitively Loaded Transmission line (CTL) will be introduced.

2.5.1 Lossless Transmission Line

A physically smooth and lossless transmission line (TL) is characterized by the

following parameters:

Characteristic Impedance:

Z0 = $ L

C Phase velocity:

up = 1" LC

= 1" m ee= constant dependingonmedium

where

L=

Z0up, C = 1

upZ0 Since eLC µε= therefore for a given dielectric constant rε , it is not possible to reduce

pu by increasing inductance or capacitance per unit length because an increase in

inductance L leads to a decrease in capacitance C. ÆL ó ∞C

Hence for a physically smooth transmission line, reduction in phase velocity pu is only

possible by increasing rε [4].

Figure 7: Lossless transmission line circuit

16

2.5.2 Capacitive Loaded Transmission Lines (CTL)

By removing the restriction that the line should be physically smooth, an effective

increase in the shunt capacitance per unit length C can be achieved without a decrease

in inductance L [4]. This is achieved by loading a transmission line with shunt

capacitance Cp at periodic intervals d.

Here, the CTL is formed by loading a microstripline with shunt capacitance created

using open stubs at periodic intervals which are much shorter than the guide

wavelength as shown in Figure 8. This causes the periodic structure to exhibit slow

wave characteristics.

d

l-stub

w_stub

C+Cp/d

L L L

C+Cp/d C+Cp/d

Figure 8: Capacitively loaded transmission line

17

The effective characteristic impedance and phase velocity of the CTL are given by the

following equations:

0 ( )CTLp

LZ CC

d

= Ω+

Where lumped capacitance per unit lengthpCd

=

-1

1

0

0

1

02

1 (ms )( )

1

1 )

pCTLp

p

p p

p

p p

uC

L Cd

CZu u Z d

Z Cu u d

−

−

=

+

= +

= +

For an N section CTL, its electrical length is given by:

00 (rad)p

CTLpCTL

CNd Nd L Cu d

ωφ ω = = +

where ω0 is the frequency of

interest.

The loaded capacitance of a unit cell is given by:

( )2 20

20

( F )CTL CTLp

CTL

Z ZC

n Z Zφ

ω−

=

The above equations show that to reduce CTL phase velocity upCTL, either one or a

combination of the following methods may be used:

1. Increase the characteristic impedance of the unloaded unit cell 0Z this is achieved

by reducing the microstripline width TLW

2. Reduce the distance between stubs d

18

3. Increase the loaded capacitance pC , this is achieved by controlling the following

stub parameters:

For an open circuit stub, 1tan[ ]

tan[ ]

stuboc

p

pstub

ZZj j C

CZ

β ω

βω

= =

=

Where pC can be increase by:

a) increasing the stub electrical length 2πβ →

b) Reducing the stub characteristic impedance stubZ↓ . This is achieved by

increasing the width of the stub stubW↑

The effects of varying these parameters can be verified with ADS circuit simulation. A

demonstration of the effects of varying CTL parameters is also shown in section 3.3.1.

19

Chapter 3 : Closed Loop Resonator Miniaturisation

3.1 Introduction

Microstrip closed loop resonators are commonly used for applications such as filters,

measurement of dielectrics, couplers, magic-T circuits and antennas. However its large

physical size can present a drawback. Hence there is strong interest to miniaturize such

resonators particularly for filter applications.

Miniaturization of microstrip filters and resonators may be achieved by using high

dielectric constant substrates or lumped elements, but very often for specified

substrates, a change in the geometry of filters is required and therefore new filter

configurations become possible. One of the ways in which resonator size can be

miniaturised without a change in substrate is by meandering the lines to create a folded

microstrip resonator [10].

Various methods that have been explored to achieve miniaturization are, meandering

the lines to create a folded microstrip resonator [10], and the use of capacitively loaded

transmission lines [1].

In the case of the folded microstrip resonator, size is reduced by meandering the lines

to form a folded ring structure. This level of miniaturization is determined by the

number of meandering sections and the tightness of the meanders. The level of

compactness achievable is however limited by parasitic coupling which occurs if

adjacent lines are located too close to each other.

In the case of the capacitively loaded microstrip loop resonator, miniaturization is

achieved by using open stubs placed at regular intervals inside the loop. The stubs

provide capacitive loading and creates a slow wave effect.

20

3.2 Novel Closed Loop Resonator

This thesis proposes a novel closed loop resonator structure that achieves both

miniaturization and improved coupling by using slow wave structures in the form of

capacitively loaded transmission lines. CTL is applied on the closed loop resonator by

placing stubs at regular intervals around the circumference. Unlike the previously

explored structures, the new resonator structure uses both inward and outward pointing

open stubs as shown in Figure 10 - Figure 12.

By using double stubs instead of single stubs, the loaded capacitance Cp can be

doubled without increasing the total size, thus further reducing the phase velocity,

resonant frequency and size. The stubs are spaced such that a stub of equivalent width

can be slotted into the space between two stubs. This enables a structure similar to an

interdigital capacitor to be formed when an identical resonator is placed in close

proximity. The effect is an increase in coupling between resonators which aids filter

synthesis.

This section demonstrates the effectiveness of resonator miniaturization using the new

structure and formulates a method of synthesizing miniaturized closed loop resonators

of a particular frequency. For standardization, the resonators shown from this point

onwards are designed on Rogers Substrate RO6010, εr=10.2, h=25 mils.

Figure 9 to Figure 13 features square closed loop and miniaturized closed loop

resonators which will be used to demonstrate resonator miniaturization and the effect

of varying CTL parameters. To determine resonator characteristics, the resonators are

weakly coupled to ports using 50Ω feed lines that are separated from the resonators by

a 10 mil gap. This keeps loading to a minimal.

21

A brief description of each of the featured resonators is as follows:

a. Resonator square closed loop shown in Figure 9 is used as a reference for

comparison against the miniaturised structures of similar size. Resonators

shown in Figure 10, Figure 11, Figure 12 are miniaturised closed loop

resonators designed with the CTL structure. These designs are such that the

total length and width is equivalent to Figure 9. This enables performance

comparison with respect to a fixed size to be made.

b. Resonator double_stub shown in Figure 10, features a simple case of a closed

loop resonator with CTL. This will be used as a reference for comparison of the

effects of varying various CTL parameters namely stub width and stub

separation.

c. Resonator dbl_w35 shown in Figure 11, features a variant of Figure 10 with

stub width increased by 50% from 23 to 35 mils. The stub width is selected

such that a gap of 10 mils between the stubs is achieved when two resonators

are placed together. This structure is created to demonstrate the effects of

increasing stub width on resonance frequency.

d. Resonator Figure 12, features a variant of Figure 10 with the number of stubs

per side increased. The distances between the stubs are selected such that a gap

of 10 mils between the stubs is achieved when two resonators are placed

together. This structure is created to demonstrate the effects of increasing the

number of stubs on resonance frequency.

e. Resonator Figure 13, features a square closed loop resonator with resonant

frequency 1.08GHz, equal to that in Figure 12.

The frequency response plot of the resonators is shown in Figure 14. A bar chart to

compare the resonance frequency is shown in Figure 15.

22

Substrate: Rogers 6010 h=25mils, εr = 10.2

833.5

833.5

Figure 9: Square closed loop f0=1.42 GHz

833.5

833.5

90

67678.75

23

Figure 10: Double_stub f0=1.19 GHz 833.5

833.5

67678.75

35

90

Figure 11: dbl_w35 f0=1.14 GHz

833.5

833.5

67678.75

66

23

Figure 12: dbl_d66 f0=1.08 GHz 1050

1050

Figure 13: Square closed loop at f0=1.08 GHz

23

m3freq=m3=-49.507

1.080GHz

m4freq=m4=-35.524

1.140GHz

m5freq=m5=-44.259

1.190GHz

m6freq=m6=-29.167

1.420GHz

m3freq=m3=-49.507

1.080GHz

m4freq=m4=-35.524

1.140GHz

m5freq=m5=-44.259

1.190GHz

m6freq=m6=-29.167

1.420GHz

0.5 1.0 1.5 2.0 2.50.0 3.0

-100

-80

-60

-40

-120

-20

freq, GHz

dB(c

lose

d_lo

op_r

eson

_mom

..S

(2,1

)

m6

dB(d

oubl

e_st

ub_m

om..

S(2

,1))

m5dB

(dbl

_w35

_mom

..S

(2,1

)) m4dB

(dbl

_d66

_mom

..S

(2,1

))

m3

M3 dbl_d66 M5 double_stub

M4 dbl_w35 M6 square closed loop Figure 14: Resonator frequency response

dbl_

d66

dbl_

w35

doub

le_s

tub

squa

re c

lose

d lo

op

1

1.1

1.2

1.3

1.4

1.5

type of resonator

reso

nanc

e fr

eq (G

Hz)

Figure 15: Compare resonance frequency

24

Resonator Miniaturization

To demonstrate resonator miniaturization, two resonators with equivalent resonant

frequency are compared.

• Miniaturized resonator shown in Figure 12 is compared against the

• Square loop resonator shown in Figure 13.

Comparing their sizes, the miniaturized resonator achieves 20% reduction in both

horizontal and vertical dimensions and a 37% reduction in area. This shows the ability

of the new structure to achieve miniaturization.

Varying CTL parameters

To demonstrate the effect of varying CTL parameters for a fixed resonator size, three

resonators shown in Figure 10 to Figure 12 which occupy the same total length and

width, are simulated using IE3D and compared.

From the simulation results in Figure 15, the effect of varying CTL parameters is

observed:

- Increasing stub width causes resonant frequency to decrease as can be seen by

comparing the resonant frequency of resonator double_stub shown in Figure 10

with dbl_w35 shown in Figure 11.

- Increasing number of stubs reduces resonant frequency as can be seen by

comparing the resonant frequency of resonator double_stub shown in Figure 10

with dbl_d66 shown in Figure 12.

The above observations correspond with the properties of CTL structures described in

2.5.

Hence the results show that the new miniaturized resonator structure enables lower

resonant frequency and hence miniaturisation to be achieved. In addition, it shows that

the level of miniaturization can be controlled by varying the unit cell parameters.

25

3.3 Resonator Synthesis Procedure

Given a desired resonator frequency f0, the procedure to generate a miniaturized closed

loop resonator is illustrated in the flow chart in Figure 20. A detailed description of the

procedure is as follows:

1) Specify the following resonator parameters:

• f0 − resonant frequency

• ZOCTL –characteristic impedance of the CTL unit cells and unloaded

microstrip lines used to form the loop.

To prevent mismatch within the loop, all unit cells and unloaded microstrip lines

should use the same characteristic impedance.

2) Specify the combination of unit cells used to form the sides of the resonator. This

involves specifying the type, number and electrical length of the unit cells used to

form each side of the resonator. For the case of a square resonator, each side has an

electrical length of φ =90°. A suggested unit cell combination consists of:

• Ndbl_stub double stub cells,

• 2 single stub cells and

• 2 unloaded transmission lines at the corner for tuning.

This combination encourages the maximization of double stub cells in the design to

enable further miniaturization. In the design, the single stub cells are placed at the

sides due to obstructions near the corners and the unloaded microstrip lines are

placed before the corners to aid fine tuning. As an example, the unit cell

combination for the miniaturized resonator introduced earlier in Figure 12 consists

of:

• 7 double stub cells of electrical length 10° (Ndbl_stub=7, Φdbl_CTL=10°)

• 2 single stub cells of electrical length 7° at the sides (Nsgl_stub=2,

Φsgl_CTL=7°)

• 2 unloaded TL at the corners of electrical length 3° (NTL=2, ΦTL=3°)

A close up view of each side is shown in Figure 18. The electrical length of each

side can be calculated using the formulae:

_ _ _ _ 90TL TL sgl stub Sgl CTL dbl stub dbl CTLN N Nφ φ φ+ + ≤

26

3) Calculate the dimensions of the single and double stub CTL unit cells.

The electrical length a CTL unit cell is given by the formulae:

00

pCTL

p

Cd d L Cu dωφ ω

= = +

.

where ω0 is the frequency of interest, L and C are related to the transmission line

characteristic impedance and loading [4].

The procedure to design a microstrip CTL is as follows:

a) First set the characteristic impedance of the unloaded unit cell Z0 and determine

the required stub capacitance Cp as follows:

0 0 ( )CTLp

L LZ Z CC Cd

= > = Ω +

( )2 2020

(F)CTL CTLp

CTL

Z ZC

Z Zφ

ω

−= .

Note that for the case of double stub unit cells, each stub can be assumed to

provide a load capacitance of Cp/2.

b) With Z0, the parameters of the unloaded microstrip line can be calculated:

WTL - width of unloaded microstrip line

up - phase velocity of unloaded microstrip line

εe - effective dielectric constant

[ ]( )

[ ]-10

1 1 12 2 1 12 /

(ms )

r re TL

TL

pe TL

wh W

cuw

ε εε

ε

+ −= +

+

=

c) Specify the desired phase velocity of the loaded CTL unit cell upCTL, and

calculate unit cell length d. -1 (ms )pCTL pu u<

(m)CTL pCTLud

ωΦ

=

27

d) Select a suitable stub width Wstub such that coupling between adjacent stubs is

minimized. A suggested distance between stubs is d-2h, where h is the

substrate thickness. Calculate the stub dimensions shown in Figure 16 and

Figure 17:

Zstub characteristic impedance of stub

Φstub electrical length

lstub physical length of stub 1tan (rad)stub p stubC Zφ ω− =

(m)stubstub

p

lC

φω

= .

e) Simulate the unit cell using EM simulation. Tune the structure to the correct

electrical length by adjusting lstub until the desired electrical length is achieved.

Store the final S-parameter file for resonator synthesis.

4) Cascade the cells to form the desired resonator. The resonator can be modelled and

simulated using ADS by using the unit cell S-parameter data and microstrip

transmission line models as shown in Figure 19.

5) Tune the resonator to the required frequency by adjusting the length of the

unloaded transmission lines lTL at the corners of the square resonator. This

parameter is chosen because it is the most predictable and easiest to modify. The

use of circuit simulation instead of EM simulation for tuning enables significant

reduction in simulation time. The final design can be verified by performing an EM

simulation and comparing with the circuit simulation results. The results are

expected to be similar if coupling between stubs is kept minimal.

28

d

l stub

wstub

WTL

Φsgl_stub= electrical length of single stub cell

Figure 16: Single stub unit cell

d

l stub

wstub

WTL

l stub

Φdbl_stub= electrical length of double stub cell

Figure 17: Double stub unit cell

Figure 18: Cascaded unit cells for a single side

29

S2P

SNP40File="ATL3_d66.s2p"

2

1

Ref

S2P


2

1

Ref

S2P


21

Ref

S2P


21

Ref

S2P


21

Ref

S2P

SNP7File="ATL_d66"

21

Ref

S2P

SNP6File="ATL_d66"

21

Ref

S2PSNP38File="ATL3_d66.s2p"

2

1

Ref

S2P


2

1

Ref

S2PSNP35

File="ATL3_d66.s2p"

2

1

Ref


2

1

Ref

S2PSNP31

File="ATL3_d66.s2p"

2

1

Ref

S2P


2

1

Ref

S2PSNP29

File="ATL3_d66.s2p"2

1

Ref


2

1

Ref

S2P


2

1

Ref

S2PSNP26


1

Ref

S2P


21

Ref

S2P


21

Ref

S2P


21

Ref

S2P


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref

S2P

SNP11File="ATL_d66"

2

1

Ref

S2PSNP10File="ATL_d66"

2

1

Ref

S2P

SNP9File="ATL_d66"

2

1

Ref


2

1

Ref


21

Ref


21

Ref

TermTerm12

Z=50 OhmNum=12

ATL3adual_stub32

W_stub=wstub

L_stub=lstubL1=L1L2=L1W=W1

2

3

1

MLINTL59

Mod=KirschningL=L0/2W=W0Subst="MSub1"

MGAPGap10

S=sepW=W0

Subst="MSub1"

MLINTL60

Mod=Kirschning

L=L2W=W0

Subst="MSub1"

MLINTL61

Mod=KirschningL=L2W=W0

Subst="MSub1"

MCORN

Corn17

W=W1

Subst="MSub1"

MLINTL62


Subst="MSub1"

MCORNCorn18

W=W1Subst="MSub1"

MLINTL63

Mod=Kirschning

L=L2W=W0

Subst="MSub1"

MCORN

Corn19

W=W1

Subst="MSub1"

MLIN

TL64

Mod=Kirschning

L=L2W=W0

Subst="MSub1"

MLIN

TL65

Mod=Kirschning

L=L2W=W0

Subst="MSub1"

MCORNCorn20

W=W1Subst="MSub1"

MLINTL66

Mod=KirschningL=L2

W=W0Subst="MSub1"

MLINTL67

Mod=KirschningL=L2

W=W0Subst="MSub1"

MLINTL68

Mod=Kirschning

L=L0/2W=W0Subst="MSub1" MGAP

Gap11

S=sepW=W0

Subst="MSub1"

ATL3adual_stub52

W_stub=wstub

L_stub=lstubL1=L1L2=L1W=W1

2

3

1TermTerm13

Z=50 Ohm

Num=13

Double stub cells

Unloaded TL

Single Stub cell

90 deg

Figure 19: Circuit model of miniaturized resonator in ADS

30

Figure 20: Resonator Synthesis Procedure

31

3.3.1 Example

To demonstrate miniaturized resonator synthesis using the method described, an

example is shown. In this example, the resonator will be simulated using various

methods to display the effects of unit cell model accuracy on the final simulated result.

Resonator specifications: f0=1.10 GHz, ZOCTL= 30 Ω;

Substrate: RO6010 ε r= 10.2 , h =25 mils.

1) The unit cell combination is chosen to be as follows:

• dual stub CTL - Ndbl_stub =7, φdbl_CTL = 10°

• single stub CTL - Nsgl_stub =2 , φsgl_CTL = 7°

• unloaded TL - NTL =2, φTL = 3°

2) The unit cell synthesis procedure is applied to determine unit cell parameters.

First, the characteristic impedance of unloaded unit cell is set: Z0 = 50 Ω > Z0CTL

=30 Ω

The required stub capacitance is calculated using ( )2 2

020

(F)CTL CTLp

CTL

Z ZC

Z Zφ

ω

−= .

For the double stub unit cell, the net capacitive loading provided by the two stubs

is calculated to be Cp=0.68pF.

Hence the capacitance provided by each stub can be assumed to be Cp_stub=0.34pF

With Z0=50 Ω the unloaded transmission line parameters are determined:

8 -1

23 (mils)

1.147 10 (ms )TL

p

W

u

=

= ×

The unit cell phase velocity is arbitrary selected as up_CTL=0.56 up

The unit cell length is calculated using = 66 (mils)CTL pCTLud

ωΦ

= .

32

Setting Wstub=23 mils, the stub parameters are calculated

Zstub = 50 Ω

_

=1.93 (mm) = 78.75 (mil)stubstub

p stub

lCφ

ω= .

Double Stub CTL

To highlight the difference in simulation results, the double stub unit cell is simulated

using both ADS circuit models shown in Figure 22 and EM simulation as shown in

Figure 21. The simulated results are shown Figure 23. The results show that the EM

simulation results (Marker m2) differs from the circuit simulation (Marker m1) by

approximately 0.61°. This equates to approximately 6% error at the desired frequency.

Hence the use of circuit models should be avoided in this example.

d = 66 mil

WTL =23 mil

Wstub=23 mil

lstub=78.75 mil

d

l stub

wstub

WTL

l stub

Figure 21: Double stub EM model

33

TermTerm11

Z=30 OhmNum=11

TermTerm10

Z=30 OhmNum=10

MCROSCros1

W4=23 milW3=23 milW2=23 milW1=23 milSubst="MSub1"

MLINTL2

Mod=KirschningL=21.5 milW=23 milSubst="MSub1"

MLINTL1


MLOCTL3


MLOCTL4


Figure 22: Double stub ADS circuit model

m1freq=m1=-9.304

1.100GHzm2freq=m2=-9.908

1.100GHzm1freq=m1=-9.304

1.100GHzm2freq=m2=-9.908

1.100GHz

0.5 1.0 1.50.0 2.0

-50

-40

-60

-30

-10

-5

-15

0

freq, GHz

dB

(S(3

,3))

ph

ase

(S(4

,3))

m1

ph

ase

(S(9

,8))

m2

dB

(S(9

,9))

double stub

Figure 23: Double stub response

34

Single Stub CTL

To enable structure symmetry, the single stub unit cells are created to be of the similar

dimension as the double stub unit cells. To highlight the difference in simulation

results, the single stub unit cell is simulated using both ADS circuit models shown in

Figure 25 and EM simulation as shown in Figure 24. The simulated results illustrated

in Figure 26 shows that the EM simulation results (Marker m2) differs from the circuit

simulation results (Marker m1) by approximately 0.8°. This equates to approximately

9% error at the desired frequency. Hence the use of circuit models for single stub unit

cells should be avoided in this example.

d = 66 mil

WTL =23 mil

Wstub=23 mil

lstub=78.75 mil

d

l stub

wstub

WTL

Figure 24: Single stub EM model

TermTerm11

Z=30 OhmNum=11

TermTerm10

Z=30 OhmNum=10

MLOCTL3


MLINTL2


MLINTL1


MTEE_ADSTee1

W3=23 milW2=23 milW1=23 milSubst="MSub1"

Figure 25: Single stub circuit model

35

m1freq=1.100GHzphase(S(2,1))=-7.471

m2freq=1.100GHzphase(S(9,8))=-8.248

0.5 1.0 1.50.0 2.0

-55

-50

-45

-40

-35

-60

-30

-10

-5

-15

0

freq, GHz

phase(S(2,1))

m1dB

(S(1

,1))

phase(S(9,8))

m2dB

(S(9

,9))

single stub

Figure 26: Single stub response

36

3) The miniaturized closed loop resonator is formed by cascading the synthesized unit

cells as shown in Figure 27. To overcome inaccuracies in the circuit models for the

unit cells, the S-parameter results from EM simulations are used to model the cells.

The resonator is simulated using circuit simulation, and the unloaded microstrip

lines at the corners are tuned to lTL =18 mils.


2

1

Ref


2

1

Ref

S2P


21

Ref

S2P


21

Ref

S2P


21

Ref


21

Ref


21

Ref

S2PSNP38


1

Ref


2

1

Ref


2

1

Ref


2

1

Ref


2

1

Ref


2

1

Ref

S2P


2

1

Ref


2

1

Ref


2

1

Ref


2

1

Ref

S2P


21

Ref

S2P


21

Ref

S2P


21

Ref

S2P


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


21

Ref


2

1

Ref


2

1

Ref


2

1

Ref


2

1

Ref


21

Ref


21

Ref

TermTerm12

Z=50 OhmNum=12

ATL3adual_stub32

W_stub=wstubL_stub=lstub

L1=L1L2=L1W=W1

2

3

1

MLINTL59

Mod=Kirschning

L=L0/2W=W0Subst="MSub1"

MGAPGap10

S=sepW=W0Subst="MSub1"

MLINTL60

Mod=KirschningL=L2W=W0Subst="MSub1"

MLINTL61


Subst="MSub1"

MCORNCorn17

W=W1Subst="MSub1"

MLINTL62


Subst="MSub1"

MCORNCorn18

W=W1Subst="MSub1"

MLINTL63


MCORNCorn19

W=W1Subst="MSub1"

MLINTL64


MLINTL65


MCORNCorn20

W=W1Subst="MSub1"

MLINTL66


MLINTL67


MLINTL68

Mod=KirschningL=L0/2W=W0Subst="MSub1" MGAP

Gap11

S=sepW=W0Subst="MSub1"

ATL3a

dual_stub52

W_stub=wstubL_stub=lstubL1=L1L2=L1

W=W1

2

3

1TermTerm13

Z=50 OhmNum=13

833.5

833.5

67678.75

66

23

Figure 27: Synthesized resonator

37

4) To compare simulation accuracy, the resonator is modelled and simulated using

three different methods namely circuit simulation, EM simulation and hybrid

circuit simulation where S-parameter model of the unit cells are used. The

simulated results are shown in Figure 28.

m1freq=1.180GHzdB(S(2,1))=-45.649

m2freq=1.080GHzdB(dbl_d66_mom..S(2

m3freq=1.100GHzdB(S(13,12))=-51.948

0.9 1.0 1.1 1.2 1.3 1.4 1.50.8 1.6

-80

-60

-40

-100

-20

freq, GHz

dB(S

(2,1

))

m1

dB(d

bl_d

66_m

om..S

(2,1

))

m2

dB(S

(13,

12)) m3

dbl_d66

Figure 28: Synthesized resonator dB|S21|

Marker m1 displays the circuit simulation results where the unit cell is modelled

using microstrip line and microstrip T-junction models in ADS.

Marker m2 displays the EM simulation results and is used as a reference.

Marker m3 displays the hybrid circuit simulation results where the unit cells are

replaced with the S-parameters from EM simulation.

The results show that the circuit simulation (Marker m1) is inaccurate and differs

significantly from the EM simulation results. This occurs due to inaccuracies in the

circuit model of the unit cell which translates to error in the predicted electrical length.

This error increases with the number of cascaded sections.

The hybrid circuit simulation (Marker m3) on the other hand provides a close

approximation and differs from the EM simulation results by only 0.02 GHz which is

less than 2% error.

38

Hence this shows that the design methodology provides a fairly effective method for

designing the miniaturized resonators.

3.4 Summary

The example has provided a simple demonstration of how a miniaturized resonator of

a desired frequency can be synthesized. To achieve further size reduction, the

following parameters can be varied:

o Increase the number of stubs Ndbl_stub

o Reduce the distance between stubs d

o Increase stub width Wstub

o Increase stub length lstub

However, these parameters cannot be increased indefinitely due to layout, coupling

and fabrication constraints. Hence achieving maximum size reduction is a constraint

optimization problem and will not be covered in this thesis.

39

Chapter 4 : Filter Synthesis Using Arbitrary Resonator Structures

4.1 Band Pass Filters

Bandpass filters may be defined by only three entities [7]; a resonator structure,

coupling K between resonators (internal coupling) and coupling to the terminations Q1

& Qn (external coupling).

A straight forward design procedure based on these concepts begins with a prototype

defined by k and q values. The values may be derived from the lowpass prototype g

values as follows

Normalized:

=

=

+

+

even n

odd n

1

1

101

n

n

nn

n

gg

ggq

ggq

, 11

1 for 1 to -1.i ii i

k i ng g+

+

= =

The k and q values are normalized to a fractional bandwidth of one.

Denormalized:

BWfqQ

BWfqQ

nn0

011

=

=

0,, f

BWkK jiji = .

The denormalized values are obtained by including a non-unity fractional bandwidth

factor.

With the denormalized K and Q values, filter synthesis can be performed in two ways:

The values can be used with analytical expressions to design a bandpass filter by

finding element values. This method is used to design top-C filters. [8]

Design of coupled resonator filters with any arbitrary resonator structure by using an

empirical method.

In this project, arbitrary resonator structures are used, hence the second method will be

elaborated in the subsequent sections.

40

4.2 Coupled Resonator Filter

There is a general technique for designing coupled resonator filters in the sense that it

can be applied to any type of resonator regardless of physical structure. It has been

applied to waveguide filters, ceramic filters and microstrip filters. This design method

is based on coupling coefficient of intercoupled resonators and external Q-factors of

the input and output resonators. [5]

4.2.1 General Coupling Matrix for Coupled Resonator Filters

This method is derived from the General Coupling Matrix, which is used to represent a

wide range of coupled-resonator filter topologies. The matrix is formulated from either

a set of loop equations or a set of node equations. This leads to a set of formulae for

analysis and synthesis of coupled-resonator filter circuits in terms of coupling

coefficient K and external Q-factors.

Figure 29: Equivalent circuit for n-coupled resonators (a) loop equation formulation, (b)

network representation

41

Representation of the network in matrix form leads to: ikjjjjjjjjjjjjjjjjjR1 + 1

¸ωC1+ ¸ωL1 ¸ωL12 ∫ ¸ωL1 n

−¸ωL21 i1 1¸ωC2

+ ¸ωL2 ∫ −¸ωL2 n

ª ª ª ª

−¸ωLn1 −¸ωL2 n ∫ Rn + 1¸ωCn

+ ¸ωLn

yzzzzzzzzzzzzzzzzz ⋅

ikjjjjjjjjj i1i2ªin

yzzzzzzzzz =

ikjjjjjjjjes0ª0

yzzzzzzzz

@ZD ⋅@iD = @eD

For simplicity, consider a synchronously tuned filter where all resonators resonate at

the same frequencyLC1

0 =ω , where nLLLL ==== …21 and

nCCCC ==== …21 . The impedance in matrix maybe expressed as @ZD = w0 ÿ L ÿ FBW ÿ@ZêêD

@ZêêD =

ikjjjjjjjjjjjjjjjjjjjj

R1w0 LÿFBW

+ p - ü w L12w0 L ÿ FBW

∫ - ü w L1 n

w0 L ÿFBW

- ü w L21w0 L ÿ FBW

p ∫ - ü w L2 n

w0 L ÿFBW

ª ª ª ª

- ü w Ln1w0 L ÿ FBW

- ü w Ln1w0 L ÿ FBW

∫ Rnw0 LÿFBW

+ p

yzzzzzzzzzzzzzzzzzzzz

Where the complex lowpass frequency variable:

p = ü 1

FBW J w

w0-

w0w

N

Note that:

Ri

w0 L= 1

Qeifor i = 1 to n

Qe1 and Qen are external Q-factors of the input and output resonators.

Define coupling coefficient

Mij =

LijL

And assuming for narrowband approximation

w

w0> 1

42

The [Z] matrix can be simplified to:

@ZêêD =

ikjjjjjjjjjjjjjjjj

1qe1

+ p - üm12 ∫ - üm1 n

- üm21 p ∫ - üm2 nª ª ª ª

- ümn1 - ümn2 ∫ 1qen

+ p

yzzzzzzzzzzzzzzzz

where

Normalized external Q factors:

qei = Qei ÿFBW for i = 1 to n Normalized Coupling coefficient:

mij =

MijFBW

By inspecting the circuit and the network, it can be identified that I1=i1, I2 = -in , and V1 = es - i1 R1 . Since

a1 = es2 "########R1

a2 = 0

b1 =es-2 i1 R12 "########R1

b2 = in è!!!!!!Rn

we have

S21 =b2a1

Àa2=0 =2 "###################R1 Rn in

es

S11 =b1a1

Àa2=0 = 1 -2 R1 i1

es Solving for i1 and in , we obtained

i1 = es

w0 LÿFBWÿ @ZêêD11

-1

in = esw0 LÿFBW

ÿ @ZêêDn1-1

Substituting the above, we have

S21 = 2"##################qe1ÿqenÿ @ZêêDn1

-1

S11 = 1 -2 R1qe1

ÿ @ZêêDn1-1

43

4.2.2 General formulation for Extracting Coupling Coefficient K

The equations above may be incorporated into a general one:

S21 = 2"##################qe1ÿqenÿ @ADn1

-1

S11 = ≤ J1 -2 R1qe1

ÿ @ADn1-1N

with [A]=[q]+p[U]-j[m]

where [q] is an n ä n matrix with all entries zero, except for q11 = 1

qe1 and qnn = 1

qen

[m] is the so called general coupling matrix, which is an n ä n reciprocal matrix (i.e.

mij = mji) and is allowed to have nonzero diagonal entries mii for an asynchronously

tuned filter.

For a given filtering characteristic of S21 and S11, the coupling matrix and the

external quality factors maybe obtained using synthesis procedures. Elements of the

matrix [m] that emerge in general will have non-zero values. For asynchronously tuned

filters, non-zero values only occur on the diagonal elements. If non-zero values lie

everywhere else, means coupling exist between every resonator, this makes it

impractical to be synthesized.

After determining the required coupling matrix for the desired filter characteristics,

next step is to establish the relationship between the value of every required coupling

coefficient and the physical structure of coupled resonators so as to find the physical

dimensions of the filter for fabrication. Couplings can be positive or negative. Positive

coupling enhances the stored energy of uncoupled resonators.

The universal formulation for extracting coupling coefficient is given by:

k = ≤ 12

J w02w01

+w01w02

N $ik w22 - w1

2

w22 + w1

2y2

-ik w02

2 - w012

w022 + w01

2y2

For synchronously tuned resonators, it simplifies to

k = ≤f22- f1

2

f22+ f1

2

44

To determine the coupling K between two resonators, the resonators are weakly

coupled to the ports and S21 is measured. The coupling coefficient is determined from

the resonant peaks of |S21| using the above formulae.

f1freq=f1=-39.402

1.365GHzf2freq=f2=-37.107

1.455GHz

1.35 1.40 1.451.30 1.50

-70

-60

-50

-40

-80

-30

freq, GHz

dB(S

(2,1

))

f1 f2

45

4.2.3 Formulation for Extracting External Quality Factor Qe

Two typical input output structures for coupled microstrip resonator filters are the

tapped line and the coupled line structure.

For tapped line coupling, usually a 50ohm feed line is directly tapped onto the

resonator and the coupling or Q-factor is controlled by the tapping position. The closer

the tapped line is to the virtual ground of the resonator, the weaker the coupling or

larger the External Q-factor.

For coupled line coupling, the feed line is linked to a coupling structure, which is

separated from the resonator by a gap. By reducing the coupling gap, a higher coupling

and hence a lower external Q can be achieved.

tapped line coupling coupled line coupling

For a singly loaded resonator, the external Q-factor can be extracted from the phase of

S11 using: 0

90sQ ω

ω±

=∆

Shown in Figure 30 is the | S11| and phase S11 of a singly loaded resonator. The

markers indicate ω+90, ω0 and ω-90. These values are substituted into the above

equation to determine singly loaded Q.

46

m7freq=m7=-84.706

1.060GHzm8freq=m8=11.783

1.010GHzm9freq=m9=-176.364

1.100GHz

0.9 1.0 1.1 1.2 1.30.8 1.4

-2

-1

0

1

-3

2

-80

20

120

-180

180

freq, GHz

dB(S

(1,1

))

phase(S(1,1))

m7

m8

m9

Figure 30: Singly loaded resonator S11

If the resonator is symmetrical, one could add another symmetrical load or port to form

a 2-port network, hence creating a doubly loaded resonator. The external Q-factor of a

doubly loaded resonator Qd is extracted from the magnitude of S21. From there the

external Q-factor singly loaded resonator Qs can be determined below.

0

3

2 2s ddB

Q Q ωω

= =∆

47

4.3 Procedure for Coupled Resonator Filter Design

The procedure for designing coupled resonator filters with arbitrary resonator

structures is briefly summarized below:

1. Compute desired filter passband Q given by BWf

Qbp0=

2. Based on the type and order of filter, determine the lowpass prototype g values

and from there calculate coefficients q & k and de-normalize the coefficients to

obtain:

nbpn

bp

qQQ

qQQ

×=

×= 11 bp

jiji Q

kK ,

, =

3. From the plot of K vs. spacing between resonators, determine the required

spacing between each resonator.

4. From the plot of SQ vs tap location, determine the port tap or coupling position.

Note:

The k and q values tabulated are based on infinite inductor QU. In practice satisfactory

results are obtainable for bpU QQ 10≥ . This method assumes that coupling between

non-adjacent resonators is not present.

48

Chapter 5 : Loaded Q and Coupling Coefficient of Resonators

5.1 Introduction

To perform coupled resonator filter synthesis, the range of achievable values for

loaded QL and coupling coefficient K of a resonator must first be determined.

The miniaturized resonator dbl_d66 synthesized in the example on section 3.3.1 is

chosen for filter synthesis (see Figure 27). This chapter begins by exploring methods

for controlling external QL. Next the range of achievable QL and K values for the

synthesized resonator are determined.

5.2 Loaded Q of Resonators

To determine the range of achievable QL, various different feed structures to the

resonator have been simulated. The feed structures are generally divided into two

categories:

1. Coupled line coupling (for high loaded Q) ;

2. Tapped line coupling (for low loaded Q) .

5.2.1 Coupled Line Coupling

For this method, the resonator is coupled by a feed line structure separated from the

resonator by a gap g. Here two different feed structures are explored:

a) Parallel coupled line (Figure 31)

b) Coupled line with interdigital stubs (Figure 32)

The parallel coupled line feed consists of a straight line located parallel to the

resonator. This feed structure is commonly used and examples of which can be found

in the following references. [1], [10].

The coupled line with interdigital stubs feed merges with the resonator dbl_d66 to

create a structure similar to an interdigital capacitor. This improves coupling and hence

increases the port loading of the structure.

For both feed structures, external QL can be controlled by varying the gap distance

between the feed structure and the resonator. The singly loaded Q is determined from

|S21| using the formulae: 0

3

2 2s ddB

Q Q ωω

′= =∆

.

49

Figure 31: Parallel coupled line feed

Figure 32: Coupled line with interdigital stubs

feed

0.00

50.00

100.00

150.00

200.00

250.00

5 10 15 20 25 30

gap

Q-lo

aded

parallel interdigital

Figure 33: Coupled line QL

The loaded Q of the structures for different gaps are simulated and compared. The

results in Figure 33, show that the Coupled line with interdigital stubs feed structure

achieves a significantly lower QL compared to the Parallel coupled line feed structure.

This proves that the interdigital structure provides a stronger coupling compared to the

parallel coupled line structure and is hence preferred due to the wider range of QL

values achievable.

50

Alternate Methods to Control Coupling of Interdigital Feed

Aside from varying the gap, the loaded Q provided by the interdigital feed can also be

controlled by:

a) Varying the number of stubs. This alters the capacitive coupling from feed to

resonator and changes QL. Shown in Figure 34 are interdigital feed structures

with varying number of stubs but with the gap fixed at 15 mils. The results in

Figure 35 show that increasing the number of stubs reduces QL due to the

increased coupling provided by the feed structure.

b) Varying the stub length of the feed to alter the capacitive coupling. This option

is currently unexplored in this thesis.

2 stubs 4 stubs

8 stubs

6 stubs

Figure 34: Coupled line with X interdigital stubs (gap=15 mils)

Q-loaded vs No. stubs

020406080

100120140160180

2 4 6 8 10

Number of stubs

Q-lo

aded

Figure 35: QL vs no. of stubs

51

5.2.2 Tapped Line Coupling

Tapped line coupling is used when low values of QL are required. The feed line usually

in the form of a 50 ohm line is directly linked to the resonator structure. QL can be

varied using two different methods:

1. Placing lumped inductors in series with the feed as shown in Figure 38.

2. Using multiple tap feed structures. This method is used to achieve precise control

of QL, and is to be applied in cases where fine control of lumped inductor L is

unavailable.

Figure 36: Tapped line coupling for dbl_d66

Figure 37: Tapped line coupling for Square

closed loop resonator

S1PSNP2File=

1 Ref

LL1

R=L=10 nH

TermTerm2

Z=50 OhmNum=2

Figure 38: Tapped line with series inductor

52

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Series Inductor L (nH)

Q-lo

aded

dbl_d66 square closed loop

Figure 39: Effect of series inductor on QL

With the use of lumped inductors in the feed, each increment to the series inductance

causes QL to be incremented accordingly as shown in Figure 39. The use of lumped

inductors provides a convenient way of adjusting QL values as it to be changed easily.

However in practice accurate control of lumped inductor values for L < 10 nH is

difficult. This is significant when accurate control of QL is required.

To achieve precise control over QL, the multiple tap feed structures shown in Figure 40

are used. By varying the number of fingers or tap points, QL can be controlled to vary

gradually over a small range (5.89 < QL< 7.78) Table 1. This is useful in applications

where fine control of QL is needed. The multiple tap feed structure can also be used

together with lumped inductors to achieve fine tuning at a higher range of QL.

53

Center stub

3 stubs

3 stubs + 2 pts

3 stubs + 4pts 5 stubs 5 stubs + 4 pts

7 stubs

5 stubs + 6 pts 7 stubs + 6pt

9 stubs 7 stubs + 8pts 9 stubs + 8pts

Figure 40: Multiple tap feed structures

Note: Detailed measurement results are shown in the reference.

54

Q-loaded (using direct tap)

5.00

5.50

6.00

6.50

7.00

7.50

8.00

cent

er s

tub

3 st

ubs

3 st

ubs

+ 2

pts

3 st

ubs

+ 4

pts

5 st

ubs

5 st

ubs

+ 4

pts

7 st

ubs

5 st

ubs

+ 6

pts

7 st

ubs

+ 6

pts

9 st

ubs

7 st

ubs

+ 8

pts

9 st

ubs

+ 8

pts

feed method

Q-lo

aded

Figure 41: QL of tapped line coupled structures

Feeding structure Q-loaded ∆Q-loaded

Center stub 5.89 -

3 stubs 5.89 -

3 stubs + 2 pts 6.00 0.11

3 stubs + 4 pts 6.19 0.3

5 stubs 6.32 0.43

5 stubs + 4 pts 6.42 0.53

7 stubs 6.69 0.8

5 stubs + 6 pts 6.77 0.88

7 stubs + 6 pts 7.00 1.11

9 stubs 7.38 1.49

7 stubs + 8 pts 7.50 1.61

9 stubs + 8 pts 7.78 1.89

Table 1: QL of tapped line coupled structures for dbl_d66

55

5.2.3 Other Explored Feed Structures

This subsection introduces other feed structures and methods that have been explored

in the process of development.

Varying feed location for Single Tap structure

Here, the effect of varying the tap location for a single tap structure on QL is explored.

The “center stub” structure in Figure 40 represents the structure with zero offset. The

structures with the shifted tap locations are shown in Figure 42.

The results displayed in Figure 43, shows that varying the tap location have little effect

on QL. Moreover the variation with tap offset position is non-linear, hence this method

is deemed to be non-effective for control of QL.

The minimal variation in QL with respect to tap position is due to the absence of forced

boundary condition such as gaps. Hence despite the shift, the ring is still analyzed as

two half wavelength resonators connected in parallel.

Up 0.5 stub length Up 1 stub length Up 1.5 stub length

Up 2 stub length Up 3 stub length Up 4 stub length Figure 42: Single tap with vertical shifted tap positions

56

5.00

5.20

5.40

5.60

5.80

6.00

6.20

6.40

6.60

6.80

7.00

0 0.5 1 1.5 2 3 4

feed tap offset position ( x stub )

Q-lo

aded

Figure 43: Single tap Q-loaded vs Offset

Varying feed location for Multiple Tap structure

The effect of shifting the feed location for the multiple tapped resonator ‘9 stub + 8pts’

shown in Figure 44 is explored here. For each of the figures, the tap location is shifted

up by X stub length from the centre line. The results in Figure 45 shows QL varies in

large steps and non-linearly with respect to offset. This is makes its shifted response

unpredictable and hence impractical to be used as a method for QL control.

Up1 stub length

Up 2 stub length

Up 3 stub length Up 4 stub length

Figure 44: Multiple tap with vertical shifted feed positions

57

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 1 2 3 4

feed tap offset position ( x stub )

Q-lo

aded

Figure 45: Multiple tap Q-loaded vs Offset

58

5.3 Coupling Coefficient K of Resonators

To determine the coupling K between closed loop resonator structures, the resonators

are placed apart at varying gap distances and weakly coupled to the ports.

The coupling coefficient is determined from |S21| using the formula:

2 2

2 12 2

2 1

f fKf f

−=

+

For resonator dbl_d66, strong coupling between the resonators is achieved by placing

the structures such that the stubs resemble an interdigital capacitor Figure 46. The

results are compared with that of the simple closed loop resonator where coupling

occurs in the form of parallel coupled lines Figure 47.

833.5

833.5

gap

Figure 46: Coupling measurement of resonator dbl_d66

833.5

833.5

gap

Figure 47: Coupling measurement of Square closed loop resonators

59

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50

resonator gap (mil)

K

dbl_d66 square closed loop

Figure 48: Coupling Coefficient K vs Gap

K gap (mil)

Square Closed loop Dbl_d66

5 0.09 0.09

10 0.06 0.08

15 0.042 0.073

20 0.032 0.068

25 0.021 0.059

30 0.011 0.055

35 Too weak 0.054

40 Too weak 0.044

45 Too weak 0.041

50 Too weak 0.032

Table 2: Coupling coefficient measurement results

Comparing the results, it is observed that:

1 For a gap > 5 mils, the coupling K achieved with resonator structure dbl_d66 in

Figure 46 is larger than that of the closed loop resonator structure Figure 47.

2 Variation of K with gap is more gradual for resonator dbl_d66 hence sensitivity to

fabrication tolerance is reduced. This enables more accurate control of K.

3 When two dbl_d66 resonators are coupled, a horizontal size reduction of one stub

length 78.75 mils is achieved due to the interdigital structure.

60

5.4 Summary

In this chapter, feed structures for the miniaturized resonator have been explored and

developed:

a) The interdigital feed structure extends the range of coupled line feed structures

and is useful in situations where high QL values are required;

b) The multiple tap feed structure provides fine control at low loaded QL values.

The coupling coefficient of the miniaturized resonator has also been measured. The

results show that new structure offers the following additional advantages over a

Square closed loop resonator of equivalent size:

• Improved control of coupling coefficient K

• Size reduction when resonators are coupled.

The range of achievable QL and K values for resonator dbl_d66 have been

characterized and the results are as follows:

Assuming a fabrication tolerance of gap ≥ 5 mils:

• Coupling coefficient: K ≤ 0.09

• QL for tapped line coupling: 5.89 ≤ QL ≤ 7.78

• QL for coupled line coupling: QL ≥ 58.9

With these results, filter synthesis can be performed.

61

Chapter 6 : Miniaturized Closed Loop Resonator Filter

6.1 Introduction

In this section filter synthesis using the miniaturized closed loop resonator dbl_d66

will be demonstrated. The synthesized filter will be compared with one that is

synthesized using the Square Closed Loop Resonator of equivalent size. Both filters

are chosen to be Chebyshev with 0.01dB ripple and 10% bandwidth.

The K & external Q values for a Chebyshev filter of 0.01dB ripple with 10%

bandwidth are shown in Table 3. Coupling Coefficient K Q-loaded

N K12 K23 K34 K45 K56 K67 K78 K89 Qin Qout

1 0.96 0.96

2 0.2337 4.489 3.7046

3 0.128 0.128 6.292 6.292

4 0.1081 0.0794 0.1081 7.129 5.883

5 0.1007 0.0697 0.0697 0.1005 7.563 7.593

6 0.097 0.066 0.0621 0.066 0.097 7.814 6.448

7 0.0949 0.0641 0.0592 0.0592 0.0641 0.0949 7.97 7.97

8 0.0936 0.063 0.0577 0.0566 0.0577 0.063 0.0936 8.073 6.6624

9 0.0928 0.0607 0.0554 0.0554 0.0554 0.057 0.0624 0.0928 8.145 8.145

Table 3: Normalized K and Q values for Chebyshev filter 0.01dB ripple 10% bandwidth

6.2 Chebyshev Filter of 0.01dB Ripple, N=3, BW=10% Using Square Closed Loop resonator

The resonator square closed loop is determined in the previous chapter to have the

following electrical characteristics:

Resonance Frequency: 1.42 GHz


Coupling coefficient: K ≤ 0.09

QL for tapped line coupling: 3.89 ≤ QL ≤ 22

.

62

The K and Q values for a third order filter N=3 are approximately translated to the

following physical parameters based on the previously simulated results in Figure 39

and Figure 48. Coupling Coefficient K Q-loaded

N K12 K23 Qin Qout

3 0.128 0.128 6.292 6.292

Gap (mil) 5 5

Feed structure

Direct tap with

4nH inductor

The synthesized filter structure is shown in Figure 49. Note that the lumped inductor is

not shown. The structure occupies an area of 2.096 in2 without the lumped inductors.

The structure is first simulated using EM simulation, next the S-parameter results are

inserted into ADS circuit simulation as shown in Figure 50 to enable the effects of

lumped inductors to be added. The final result plot in Figure 51 shows that the

simulated results are reasonably close to that desired with the bandwidth deviating by

approximately 3%.

833.5

2515.5

Figure 49: Layout of the 3rd order Chebyshev filter using Square closed loop resonator

63

TermTerm1

Z=50Num=1

TermTerm2

Z=50Num=2

LL9

R=L=4 nH

LL10

R=L=4 nH

S2Psimple_closed_loop

21

Ref

Figure 50: Circuit simulation of 3rd order Chebyshev filter using Square closed loop resonator with

series inductors placed at each port to increase QL.

s1freq=1.400GHzdB(S(2,1))=-1.202

s2freq=1.300GHzdB(S(2,1))=-4.820

s3freq=1.480GHzdB(S(2,1))=-4.195

1.2 1.4 1.6 1.81.0 2.0

-50

-40

-30

-20

-10

0

-60

10

freq, GHz

dB(S

(2,1

))

s1s2 s3

dB(S

(1,1

))

3rd order Chebyshev filter using Simple Closed Loop

Simulated Response using IE3D & ADS circuit simulation

F0 = 1.4 GHz ( |S21| = -1.47 dB)

F-3dB = 1.3 GHz ( |S21| = -4 dB )

F+3dB = 1.48 GHz ( |S21| = -4.4 dB )

BW = 13% Figure 51: Simulated response of the 3rd order filter using Square closed loop resonator

64

6.3 Chebyshev Filter of 0.01dB ripple, BW=10% Using Resonator dbl_d66.

The resonator dbl_d66 is determined in the previous chapters to have the following

electrical characteristics:

Resonance Frequency: 1.08 GHz


Coupling coefficient: K ≤ 0.09

QL for tapped line coupling: 5.89 ≤ QL ≤ 36

QL for coupled line coupling: QL ≥ 46.3

The characteristics of resonator dbl_d66, enables filter of order N ≥3 to be synthesized.

For ease of fabrication within in house facilities, a 3rd order filter is chosen to be made.

The smaller overall area enables small gaps of 5mils on different parts of the structure

to be fabricated and realized.

The K and Q values for N=3 are approximately translated to the following physical

parameters based on the previously simulated results in Figure 41and Figure 48. Coupling Coefficient K Q-loaded

N K12 K23 Qin Qin

3 0.128 0.128 6.292 6.292

Gap (mil) 5 5

Feed structure 5 stub 5 stub

Note: Gap of 5 mils rather than 3 mils is chosen for K12 & K23 to keep the design

within fabrication tolerances.

The synthesized filter shown in Figure 52 occupies an area of 2058 in2. Which is 38 in2

smaller compared to the square loop resonator filter formed using resonators of

equivalent overall size.

The simulated filter response shown in Figure 53 corresponds closely to desired result

with bandwidth exceeding by only 1%.

65

2376

866.5

Figure 52: Layout of the 3rd Order Chebyshev filter using dbl_d66

d1freq=1.080GHzdB(S(4,3))=-1.483

d2freq=1.020GHzdB(S(4,3))=-4.009

d3freq=1.140GHzdB(S(4,3))=-4.422

0.9 1.0 1.1 1.2 1.3 1.40.8 1.5

-30

-20

-10

0

-40

10

freq, GHz

dB(S

(3,3

))dB

(S(4

,3))

d1d2 d3

3rd order Chebyshev filter using dbl_d66

F0 = 1.09 GHz ( |S21| = -1.47 dB),

F-3dB = 1.02 GHz ( |S21| = -4 dB ); F+3dB = 1.14 GHz ( |S21| = -4.4 dB )

BW = 11% Figure 53: Simulated response (IE3D) of the 3rd order filter using dbl_d66

66

6.4 Fabricated and Measured Results

Illustrated in this section is the fabricated 3rd order Chebyshev filter using dbl_d66

and its measurement result. The results Figure 57 show that the measured response

corresponds closely to the simulated response. Hence filter synthesis using resonator

dbl_d66 has been successfully achieved.

Figure 54: Fabricated 3rd order filter using dbl_d66

Figure 55: VNA measurement from 50 MHz to 2400 MHz

67

Figure 56: VNA measurement 50 MHz to 1600 MHz

m1freq=1.080GHzdB(S(2,1))=-1.505

m2freq=1.005GHzdB(S(4,3))=-4.683

m3freq=1.155GHzdB(S(4,3))=-4.526

m1freq=1.080GHzdB(S(2,1))=-1.505

m2freq=1.005GHzdB(S(4,3))=-4.683

m3freq=1.155GHzdB(S(4,3))=-4.526

0.4 0.6 0.8 1.0 1.2 1.4 1.60.2 1.8

-70

-60

-50

-40

-30

-20

-10

-80

0

freq, GHz

dB(S

(2,1

))

m1

dB(S

(4,3

))

m2 m3

Measured ResultSimulated Result

Figure 57: Compare simulated vs measured result

Measured Response:

F0 = 1.08 GHz ( |S21| = -1.12 dB)

F-3dB = 1.01 GHz ( |S21| = -4.15 dB )

F+3dB = 1.15 GHz ( |S21| = -4.12 dB )

BW = 13%

68

6.5 Summary

In this chapter filter synthesis using the miniaturized resonator has been successfully

demonstrated. The measured result of the fabricated filter corresponds closely to the

simulated results as shown in Figure 54.

Comparing the size and performance of a filter created using the simple closed loop

resonator (Figure 53) against one created using the miniaturized resonator dbl_d66

(Figure 51), the following is observed:

• Filter using dbl_d66 occupies a slightly smaller area of 2.058 in2 whereas the filter

using the simple loop resonator occupies a slightly larger area of 2.096 in2. This

equates to approximately 6% reduction in area.

• Filter using dbl_d66 has a lower resonant frequency of 1.08 GHz as compared to

1.4 GHz. This equates to a 22% reduction in resonant frequency.

69

Chapter 7 : Conclusion

In this thesis, the objective of closed loop resonator miniaturization using capacitively

loaded transmission lines (CTL) has been successfully achieved. The miniaturized

resonator has been demonstrated to achieve 37% reduction in area over the square

closed loop resonator of equivalent resonant frequency. A method to synthesize

miniaturized closed loop resonators of the desired resonant frequency has been

developed.

The miniaturized resonators offer improved coupling performance and control through

the interdigital capacitor like structures that are formed when two resonators are placed

adjacent to each other.

Planar feed structures to control the external QL of the miniaturized resonators have

also been developed. The structures designed enable fine tuning of QL without the use

of discrete lumped inductors.

Filter synthesis has successfully been performed on the newly developed structure. A

3rd order Chebyshev filter of 0.01dB ripple and 10% bandwidth has been fabricated

and measured with simulation results corresponding closely to the measured results. In

comparison to a filter synthesized using a closed loop resonator of similar size, the

new structure achieves a 22% lower resonant frequency and also an area additional

area reduction of 6% which is achieved due to the interdigital structure.

70

7.1 Suggestion for Future Works

In this thesis, the application of slow wave structures, in the form of capacitive open

circuited microstrip line stubs to the closed loop resonator, has successfully enabled

resonator miniaturization to be achieved.

For future studies, other forms of capacitive loading structures could be explored. This

could be in form of lumped capacitors or other microstrip structures such as radial

stubs.

71

Chapter 8 : Appendix

Parallel Coupled Line Feed

Feed gap (mil) F1 (GHz) F2 (GHz) F0

(GHz) Q-

loaded 5 1.092 1.108 1.100 137.50 10 1.099 1.113 1.106 158.00 15 1.101 1.114 1.107 170.31 20 1.102 1.114 1.108 184.67 25 1.1026 1.114 1.108 194.39 30 1.103 1.114 1.108 201.42

72

Coupled Line with Interdigital stub feed

Reson sep (mil) F1 (GHz) F2 (GHz) F0 (GHz) Q-loaded

5 1.073 1.11 1.090 58.92

10 1.078 1.11 1.095 68.44

15 1.081 1.11 1.095 75.52

20 1.084 1.108 1.095 91.25

25 1.086 1.107 1.095 104.29

30 1.087 1.107 1.095 109.50

73

Coupled Line with X interdigital stubs (gap=15 mils)

2 stubs 4 stubs

8 stubs

6 stubs

# of stubs F1 (GHz) F2 (GHz) F0 (GHz) Q-loaded

2 1.093 1.107 1.1 157.14

4 1.085 1.107 1.095 99.55

6 1.082 1.11 1.095 78.21

8 1.081 1.11 1.095 75.52

10 1.081 1.11 1.095 75.52

74

Tapped line coupled structures

Show S-parameter Plot beside each structure.

Center stub

3 stubs

3 stubs + 2 pts

5 stubs

7 stubs

75

3 stubs + 4pts

5 stubs + 4 pts

5 stubs + 6 pts

7 stubs + 6pt

9 stubs

76

7 stubs + 8pts

9 stubs + 8pts

Feed F+90 (GHz) F-90 (GHz) F0 (GHz) Q-loaded

Center Stub 1.02 1.21 1.12 5.89

3 stubs 0.995 1.18 1.09 5.89

5 stubs 0.985 1.155 1.075 6.32

7 stubs 0.985 1.145 1.07 6.69

9 stubs 0.995 1.14 1.07 7.38

3stubs + 2pt 0.99 1.17 1.08 6.00

3stubs + 4pt 0.975 1.147 1.065 6.19

5stubs + 4pt 0.975 1.14 1.06 6.42

5stubs + 6pt 0.97 1.125 1.05 6.77

7stubs + 6pt 0.975 1.125 1.05 7.00

7stubs + 8pt 0.98 1.12 1.05 7.50

9stubs + 8pt 0.98 1.115 1.05 7.78

77

Coupling Measurement of Resonator dbl_d66

833.5

833.5

gap

78

Reson gap (mil) F1 (GHz) F2 (GHz) F0 (GHz) k

3 1.04 1.15 1.094 0.10

5 1.04 1.14 1.089 0.09

10 1.05 1.14 1.094 0.08

15 1.055 1.135 1.094 0.073

20 1.06 1.135 1.097 0.068

25 1.065 1.13 1.097 0.059

30 1.07 1.13 1.100 0.055

35 1.075 1.135 1.105 0.054

40 1.077 1.125 1.101 0.044

45 1.08 1.125 1.102 0.041

50 1.085 1.12 1.102 0.032

79

Coupling Measurement of Square Closed Loop Resonator

833.5

833.5

gap

Reson sep (mil) F1 (GHz) F2 (GHz) F0 (GHz) k

5 1.335 1.465 1.398 0.09

10 1.365 1.455 1.409 0.06

15 1.385 1.445 1.415 0.042

20 1.395 1.44 1.417 0.032

25 1.4 1.43 1.415 0.021

30 1.41 1.425 1.417 0.011

35 1.42 - 1.42 coupling too weak




80

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miniaturized loop resonator filter using …chapter 2 : microstrip resonators and slow wave...

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